164 STATICS. [269. 



Rotating the whole figure about AB as axis, the ellipses describe con- 

 focal ellipsoids of revolution which are level, or equipotential, surfaces. 



269. Exercises. 



(1) A segment AB is cut out of an infinite straight line along which 

 mass is distributed uniformly. If the mass on the ray issuing from A be- 

 repulsive, that on the ray issuing from B (in the opposite sense), attrac- 

 tive, determine the resultant attraction at any point P by the method of 

 Art. 267, and show that the lines of force are confocal ellipses, while the 

 equipotential surfaces are confocal hyperboloids. 



(2) Three rods of constant density form a triangle. Find the point 

 at which the resultant attraction is zero. 



(3) Find the attraction of a straight rod AB of constant density on 

 a point /'situated on the line AB so that AP= a, J3P = b. 



(4) Two rods of lengths 2 a, ib, and of equal constant density, are 

 placed parallel to each other, at a distance c, so that the line joining 

 their middle points is at right angles to them. Find their mutual attrac- 

 tion, i.e. the force required to keep them apart. 



(5) Show that the attraction of a homogeneous rod of infinite length 

 on a point at the distance p from it, is 2 *p/p. 



270. The formula of the last exercise (5) can be used to determine 

 the attraction of an infinitely long homogeneous cylinder of finite cross- 

 section on an external point P, by resolving the cylinder into filaments 



N 



Fig. 80. 



parallel to the axis. Fig. 80 represents the cross-section of the cylinder 

 passing through P. The polar element of area at Q, rdd dr, can be 

 regarded as the cross-section of a filament, whose attraction at P is, by 



Ex. (5), 



rdrdB j ,/\ 



2 Kp = 2 Kparati. 



